Main source: NS Ch 6 (parts)
NS: Ch 6: Production
Production function (at least the basic idea)
Returns to scale (!)
Basic cost concepts (!)
‘Cost minimising input choice’ (understand key principle of ‘bang-for-the-buck’ equalisation)
Cost curves (!)
… to help understand the key concepts in following chunks
Key goals of this chunk
Think about: How does the firm’s production choice differ from a consumer’s consumption choice?
Who cares about the firm’s/firms’/national/global production function?
Urgent question: Brexit
Trade with Europe may default to WTO terms
\(\rightarrow\) Very large tariffs on some goods, ‘non-tariff barriers’ on others
UK (and EU) firms: Unknown impact on input prices (as well as demand curves, competition, etc.)
\(\rightarrow\) Consumer prices? Wages? Investment returns?
How do we define it? What does it do?
What’s its goal?
Production function: mathematical relationship between inputs and outputs.
\[q = f( K, L, M, ...)\]
\[q = f(K, L)\]
e.g., \(q =2L^{1/3}K^{2/3}\)
Given choice of ‘amount to produce’, firm tries to produce it at minimum cost
(like consumer maximizing utility st budget constraints)
Marginal Product of Labour—\(MP_L\): Slope of production function in units of labour (holding capital etc constant)
\[MP_L=\frac{\partial}{\partial L}f(K,L)\]
Similarly for Marginal Product of Capital (\(MP_K\))
Similar to consumer’s indifference curves, but quantity is observable
RTS = - slope of the isoquant
|(change in capital) / (change in labor)|
\[RTS = MP_L/MP_K\]
Makes sense:
If MPL large I can give up much K, bc gaining L adds a lot of production
If MPK large I can’t give up much K to get more L, bc reducing K reduces output a lot
Diminishing RTS
Which combination will the firm choose?
\[q = f(K,L,...)\]
Main point: Whatever q it wants to produce, firm uses the minimum cost combination of inputs!
…chooses inputs to get the best ‘bang for the buck’;
\(\rightarrow\) where input mix optimal, each input yields same marginal product per £
Consider a combo of K and L, and the output this yields.
From here, firm can ‘substitute capital for labour’ at rate \(RTS(K,L)= MP_L/MP_K\) (& hold production constant)
If firm uses both K & L and chooses optimally \(\rightarrow\)
Set \(RTS(K,L)\) equal to the input price ratio of these inputs (\(w/r\))
\(\rightarrow\) ‘Same bang for the buck at optimal choices \(K^*\) and \(L^*\)’ i.e.,
\[\frac{MP_K(K^*,L^*)}{r}= \frac{MP_L(K^*,L^*)}{w}\]
Aside, for intuition and story-telling
If markets for labour and capital are (perfectly) competitive prices of inputs & outputs adjust so that:
‘bang for the buck’ (in revenue) is the same for all inputs and for all production processes
Inputs (workers, owners of capital) in every industry will be paid based on their (marginal) productivity …
Back to our ‘replaced by AI’ motivation, a simple model. Suppose:
One product in the economy with ‘constant returns to scale Cobb-Douglas production function:’
\[q = L^{a}K^{1-a}\]
This implies ‘optimising firms will spend a share \(\alpha\) on labour’.
Summing up: Optimisation (given a production function and input prices)…
yields a (minimum) cost for every output \(q\) a firm chooses to produce.
Are bigger firms always more efficient? Do things get cheaper to produce the more we produce?
E.g., doubling all inputs (labor, capital, land, etc) means exactly doubling all outputs
IRS
Fixed costs (incorporation, buildings, management, planning, R&D) spread over more units
Should always be able to at least ‘double everything’ and produce twice as much? (so at least CRS)
Scale allows specialisation
Arguments for DRS
Here MES is 100 \(\rightarrow\) You should produce in multiples of 100, but could be CRS for each multiple of 100
Computing… If you know the production function, how do you know if the ‘returns to scale’ are increasing or decreasing?
Slide in a constant \(\alpha>1\) next to each input, simplify, and compare to the original production
E.g.:
\[Q(L,K) = L^{1/4}K^{1/2}\]
\[Q(\alpha L, \alpha K) = (\alpha L)^{1/4}(\alpha K)^{1/2}\] \[=\alpha^{1/4}\alpha^{1/2}L^{1/4}K^{1/2}=\alpha^{3/4}L^{1/4}K^{1/2}=\alpha^{3/4}Q(L,K)\]
\(\rightarrow\) So if we increase inputs by \(\alpha\) here, we increase output by \(\alpha^{3/4}<\alpha\), so DRS everywhere for this production function.
Main source: NS Ch 7 (just a few parts)
Key goals of this chunk
Characterise and contrast different types of costs; how they should enter into firms’ decisions (and life decisions!)
Continue to depict a firm’s cost-minimising input choice
Fixed costs (FC): Costs that must be regularly incurred to remain in business (i.e., for any level of output), but that do not vary with the level of output
Variable costs (VC): Costs that increase with the quantity produced.
Sunk costs: Costs that have been incurred in the past that can never be recovered.
Sunk costs should not enter into any economic decisions.
FC from previous years are sunk costs; FC for future years are not.
Total costs \(= TC = wL + vK\)
Economic profit = \(\pi\) = Total Revenues - Total Costs
\[\pi = Pq-wL-vK = P\times f(K,L)-wL-vK\]
(P: price of good)
Choose point where RTS = ratio of input prices
Which point on this curve will minimize cost? . . . The one on the lowest ‘isocost line’ … similar to budget line for consumers
Intuition
\[RTS = MP_L/MP_K = w/v\]
Cost-minimisation for each level of output \(\rightarrow\) firm’s expansion path
Average and marginal cost curves
… with differing returns to scale: Increasing, decreasing, constant, optimal scale
Skip: “Short and long run” costs for the firm
\[AC=TC/q\]
Shape of marginal cost curve depends on production function
Constant returns to scale: constant MC (and no FC)
Decreasing returns to scale: increasing MC
Increasing returns to scale: decreasing MC (and/or constant FC)
Want to estimate product, firm, and industry production functions
Government (regulators etc), forecasters, strategists
Anti-trust, regulating natural monopolies, macro-growth issues and aggregate production, importance of human-K, impact of trade deals (winners/losers), impact of min. wage and labour laws, business strategy and competition, reacting to anticipated market changes…
Production functions \(\Leftrightarrow\) cost functions